Question

Let \((a, b)\) be a fixed point in \(\mathbb{R}^{2}\) and let \(d=f(x, y)\) be the distance between \((a, b)\) and an arbitrary point \((x, y)\) a. Show that the graph of \(f\) is a cone. b. Show that the gradient of \(f\) at any point other than \((a, b)\) is a unit vector. c. Interpret the direction and magnitude of \(\nabla f\)

Short answer

Question: Prove that the graph of the function that calculates the distance between a fixed point (a, b) and any other point (x, y) represents a cone. Additionally, show that the gradient of the function is a unit vector at any point other than (a, b) and interpret the direction and magnitude of the gradient. Answer: By calculating the distance function f(x, y) = √((x-a)² + (y-b)²) and its gradient (∇f(x, y)), we find that the squared distance function represents a cone in ℝ³ with its vertex at point (a, b, 0). Additionally, the gradient of the function at any point other than (a, b) is a unit vector with magnitude 1. The direction of the gradient vector points away from the vertex of the cone (the fixed point (a, b)) and indicates the direction in which the distance between points (a, b) and (x, y) increases the most. The constant magnitude of the gradient vector implies a constant rate of change in the distance in this direction.

Step-by-step solution

Find the expression for the function f(x, y)

The distance between two points in \(\mathbb{R}^2\) can be calculated using the distance formula. For our fixed point \((a, b)\) and an arbitrary point \((x, y)\), the distance is given by: \[f(x, y) = \sqrt{(x-a)^2 + (y-b)^2}\]

Calculate the gradient of f(x, y)

The gradient is a vector of partial derivatives with respect to each variable. In this case, we will find the partial derivatives of the function \(f(x, y)\) with respect to \(x\) and \(y\). The partial derivative of \(f(x, y)\) with respect to \(x\): \[\frac{\partial f(x, y)}{\partial x} = \frac{(x-a)}{\sqrt{(x-a)^2 + (y-b)^2}}\] The partial derivative of \(f(x, y)\) with respect to \(y\): \[\frac{\partial f(x, y)}{\partial y} = \frac{(y-b)}{\sqrt{(x-a)^2 + (y-b)^2}}\] Thus, the gradient of \(f(x, y)\) is: \[\nabla f(x, y) = \left(\frac{(x-a)}{\sqrt{(x-a)^2 + (y-b)^2}}, \frac{(y-b)}{\sqrt{(x-a)^2 + (y-b)^2}}\right)\]

Show that the graph of f is a cone

To show that the graph of \(f(x, y)\) is a cone, we will eliminate the square root in the expression by squaring the function: \[(f(x, y))^2 = (x-a)^2 + (y-b)^2\] This equation represents a cone in \(\mathbb{R}^3\) with its vertex at point \((a, b, 0)\).

Show that the gradient of f at any point other than (a, b) is a unit vector

A vector is a unit vector if its magnitude is equal to 1. The magnitude of the gradient vector can be calculated using the Pythagorean theorem: \[\|\nabla f(x, y)\| = \sqrt{\left(\frac{(x-a)}{\sqrt{(x-a)^2 + (y-b)^2}}\right)^2 + \left(\frac{(y-b)}{\sqrt{(x-a)^2 + (y-b)^2}}\right)^2}\] After simplifying, we have: \[\|\nabla f(x, y) \| = \frac{\sqrt{(x-a)^2+(y-b)^2}}{\sqrt{(x-a)^2+(y-b)^2}} = 1\] Thus, the gradient of \(f(x, y)\) at any point other than \((a, b)\) is a unit vector.

Interpret the direction and magnitude of the gradient

The gradient of a scalar function gives the direction of the maximum increase of the function and its magnitude represents the rate of change in that direction. In this case, since the gradient vector always has a magnitude of 1 (unit vector) at any point other than \((a, b)\), it means that the rate of change in the distance \(f(x, y)\) is constant in the direction of the gradient. Furthermore, the direction of the gradient vector indicates that it points away from the vertex of the cone (the fixed point \((a, b)\)), in the direction of the point \((x, y)\). In other words, the gradient vector shows the direction in which the distance between the points \((a, b)\) and \((x, y)\) increases the most.

Key concepts

Distance formula

The distance formula is a fundamental concept in geometry used to determine the distance between two points in a coordinate system, such as \[(a, b)\] and \[(x, y)\] in a 2D space. It is derived from the Pythagorean theorem and is expressed as follows:
\[f(x, y) = \sqrt{(x-a)^2 + (y-b)^2}\]
By using this formula, we can easily calculate how far apart points are from each other. When applied, it provides the straight-line or Euclidean distance, useful in various applications such as physics, engineering, and navigation.

• The formula helps determine paths and closest routes.
• It is essential for graphing equations of circles and other geometric shapes.

Understanding the distance formula allows us to explore deeper geometrical concepts, including the nature of conic sections.

Conic sections

Conic sections represent the curves obtained by intersecting a plane with a cone, resulting in shapes like circles, ellipses, parabolas, and hyperbolas. In our context, illustrating a 3D cone using the function \[(f(x, y))^2 = (x-a)^2 + (y-b)^2\] showcases the graph in three-dimensional space.

• This particular equation represents a circular cone centered at \[(a, b)\] on the plane.
• The vertex of the cone is at \[(a, b, 0)\] in 3D space.

Conic sections arise in various fields:

• Astronomy: Orbits of planets and comets take the shape of ellipses, a type of conic section.
• Architecture: Parabolic arches and hyperbolic structures are studied for strength and aesthetics.

By mastering conic sections, one can understand the geometric relationships between different shapes and their real-world applications.

Partial derivatives

Partial derivatives help us understand how a function changes as each variable changes, focusing on one variable at a time while keeping others constant. For the function \[f(x, y) = \sqrt{(x-a)^2 + (y-b)^2}\], the partial derivatives with respect to \[x\] and \[y\] are calculated to form the gradient vector:

• The partial derivative with respect to \[x\]: \[\frac{\partial f}{\partial x} = \frac{(x-a)}{\sqrt{(x-a)^2 + (y-b)^2}}\]
• The partial derivative with respect to \[y\]: \[\frac{\partial f}{\partial y} = \frac{(y-b)}{\sqrt{(x-a)^2 + (y-b)^2}}\]

These partial derivatives combine to form the gradient \[abla f\], indicating the direction of greatest increase of the function.

• The gradient vector points toward the direction in which the function's value rises the fastest.
• In this specific scenario, the gradient's magnitude remains 1, making it a unit vector that provides consistency in directional change.

Partial derivatives and gradients are crucial in multivariable calculus, optimizing functions, and analyzing vector fields.